Connecting spectral and spring methods for manifold learning

Shannon M. Hughes, Peter Jeffrey Ramadge

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations

Abstract

Diffusion Maps (DiffMaps) has recently provided a general framework that unites many other spectral manifold learning algorithms, including Laplacian Eigenmaps, and it has become one of the most successful and popular frameworks for manifold learning to date. However, Diffusion Maps still often creates unnecessary distortions, and its performance varies widely in response to parameter value changes. In this paper, we draw a previously unnoticed connection between DiffMaps and spring-motivated methods. We show that DiffMaps has a physical interpretation: it finds the arrangement of high-dimensional objects in low-dimensional space that minimizes the elastic energy of a particular spring network. Within this interpretation, we recognize the root cause of a variety of problems that are commonly observed in the Diffusion Maps output, including sensitivity to user-specified parameters, sensitivity to sampling density, and distortion of boundaries. We then show how to exploit the connection between Diffusion Map and spring criteria to create a method that can be efficiently applied post hoc to alleviate these commonly observed deficiencies in the Diffusion Maps output.

Original languageEnglish (US)
Title of host publication2009 IEEE International Conference on Acoustics, Speech, and Signal Processing - Proceedings, ICASSP 2009
Pages1565-1568
Number of pages4
DOIs
StatePublished - Sep 23 2009
Event2009 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2009 - Taipei, Taiwan, Province of China
Duration: Apr 19 2009Apr 24 2009

Publication series

NameICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings
ISSN (Print)1520-6149

Other

Other2009 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2009
CountryTaiwan, Province of China
CityTaipei
Period4/19/094/24/09

All Science Journal Classification (ASJC) codes

  • Software
  • Signal Processing
  • Electrical and Electronic Engineering

Keywords

  • Multidimensional signal processing
  • Unsupervised learning

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